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1 Adversarial Search Chapter 6 Section 1 – 4 The Master vs Machine: A Video

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2 Outline Optimal decisions α-β pruning Imperfect, real-time decisions

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3 Games vs. search problems "Unpredictable" opponent specifying a move for every possible opponent reply Time limits unlikely to find goal, must approximate Example: tic-tac-toetic-tac-toe –Relationship to five-in-a-row?

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4 Game tree (2-player, deterministic, turns)

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5 Minimax Perfect play for deterministic games Idea: choose move to position with highest minimax value = best achievable payoff against best play E.g., 2-ply game: MAX MIN

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6 Minimax algorithm

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7 Another Example (From Nilsson “Principles of AI”)

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8 Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent) Time complexity? O(b m ) Space complexity? O(bm) (depth-first exploration) For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible

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9 α-β pruning example

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10 α-β pruning example

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11 α-β pruning example

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12 α-β pruning example

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13 α-β pruning example

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14 Rules for alpha and beta IF –node=MIN and ancestor=MAX –beta(node) <= alpha(ancestor), –then stop this branch IF –node=MAX and –alpha (node) >= beta(parent), –then stop this branch

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15 More examples

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16 Properties of α-β Pruning does not affect final result Good move ordering improves effectiveness of pruning With "perfect ordering," time complexity = O(b m/2 ) doubles depth of search A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)

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17 Why is it called α-β? α is the value of the best (i.e., highest- value) choice found so far at any choice point along the path for max If v is worse than α, max will avoid it prune that branch Define β similarly for min

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18 The α-β algorithm

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19 The α-β algorithm

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20 Resource limits Suppose we have 100 secs, explore 10 4 nodes/sec 10 6 nodes per move Standard approach: cutoff test: e.g., depth limit (perhaps add quiescence search) evaluation function = estimated desirability of position

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21 Evaluation functions For chess, typically linear weighted sum of features Eval(s) = w 1 f 1 (s) + w 2 f 2 (s) + … + w n f n (s) e.g., w 1 = 9 with f 1 (s) = (number of white queens) – (number of black queens), etc.

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22 Cutting off search MinimaxCutoff is identical to MinimaxValue except 1.Terminal? is replaced by Cutoff? 2.Utility is replaced by Eval Does it work in practice? b m = 10 6, b=35 m=4 4-ply lookahead is a hopeless chess player! –4-ply ≈ human novice –8-ply ≈ typical PC, human master –12-ply ≈ Deep Blue, Kasparov

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23 Deterministic games in practice Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions. Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply. –http://www.math.duke.edu/~blake/chess.htmlhttp://www.math.duke.edu/~blake/chess.html Othello: human champions refuse to compete against computers, who are too good. Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

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24 Summary Games are fun to work on! They illustrate several important points about AI perfection is unattainable must approximate good idea to think about what to think about

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